Role of Parameter Errors in the Spring Predictability Barrier for ENSO Events in the Zebiak-Cane Model

The El Ni#cod#241;-Southern Oscillation (ENSO) cycle, the most prominent year-to-year climate variation on the Earth, has a large impact on climate and weather events. Therefore, an understanding of this phenomenon is crucial for scientists all over the world (Wyrtki, 1975; Suarez and Schopf, 1988; Neelin, 1991; Wang and Picaut, 2004; McPhaden et al., 2006).

Because of the complexity and nonlinearity of this phenomenon (An and Jin, 2004; Rodgers et al., 2004), researchers have recognized the shortcomings of theoretical models, and thus certain sophisticated models have been developed and broadly used to study it (Neelin, 1990; Kleeman, 1991; Penland and Magorian, 1993; Luo et al., 2008). The Zebiak-Cane model (ZC model) (Zebiak and Cane, 1987) addressed in the present paper is one such model——a nonlinear anomaly model of intermediate complexity. The model is composed of a Gill-type steady-state linear atmospheric model and a reduced-gravity oceanic model, which has been highly important for researchers in understanding and predicting ENSO events (Zebiak and Cane, 1987; Blumenthal, 1991; Xue et al., 1994).

The spring predictability barrier (SPB) is one of the big challenges for scientists involved in studying the prediction of ENSO. The SPB refers to the fact that, when forecasts are built before and across the boreal spring season, the forecast skill tends to show a significant drop during this season (Yu and Kao, 2007). Many researchers have studied the SPB from the perspective of initial error growth and have obtained certain meaningful results. (Moore and Kleeman, 1996) investigated the growth of a linear singular vector (LSV) to study the SPB. They found that the LSV has the largest error growth tendency in boreal spring. Chen et al. (1995, 2004) noted that the SPB could be reduced by improving the initial input of the model. They improved ENSO prediction skill in the ZC model by data assimilation. (Mu et al., 2007) used the ZC model and the conditional nonlinear optimal perturbation (CNOP) approach to study the SPB and calculated the optimal initial errors (CNOP-I errors) under the El Ni#cod#241; background. This group noted that the growth of the CNOP-I errors depends on seasons, and that these errors can lead to the largest prediction error and a noticeable SPB. They also showed that the spatial structure of the initial errors is highly important in the generation of the SPB. (Duan et al., 2009b) and (Yu et al., 2009) identified two types of CNOP-I errors leading to the SPB and described two different methods of error growth. Additionally, (Yu et al., 2009) indicated that the CNOP-I errors of the sea surface temperature anomaly have a dipolar spatial structure, which is more important to the SPB than the random initial errors that lack a particular spatial structure. Furthermore, (Duan and Wei, 2012) demonstrated that CNOP-like errors (errors that are similar to CNOP errors) exist in the analysis fields of ENSO hindcasts generated by the Flexible Global Ocean Atmosphere Land System Model-gamil (Grid-point Atmospheric Model of IAP/LASG), i.e. FGOALS-g model (Yan and Yu, 2012), which indicates that the forecasting ability of ENSO will be greatly improved if CNOP-like errors are filtered from the analysis fields.

In addition to the initial errors, parameter errors are another cause of prediction errors. (Liu, 2002) chose different parameter errors to directly study how the uncertainty of the parameter errors impacts upon ENSO simulation and the extent to which ENSO is dependent on and sensitive to the uncertainty of the parameter errors. (Zebiak and Cane, 1987) investigated the simulation of the ZC model with parameter perturbation. All parameter changes that amounted to increasing (decreasing) the strength of the atmosphere-ocean coupling tended to produce larger (smaller) amplitudes and longer (shorter) periods. (Macmynowski and Tziperman, 2008) studied the sensitivity of the ENSO period to different parameters, and found that the period of ENSO depends on three physical processes. (Mu et al., 2010) extended the CNOP approach and used this new method to study the parameter errors. This group noted that the optimal parameter errors (CNOP-P errors) are less important than the optimal initial errors (CNOP-I errors) for the overall prediction errors in a theoretical coupled ocean-atmosphere model. Using the WF 96 model (Wang and Fang, 1996), (Duan and Zhang, 2010) showed that the CNOP-P errors lead to neither a noticeable prediction error nor a significant SPB. (Yu et al., 2012) demonstrated a result similar to that of (Duan and Zhang, 2010) with the ZC model and the CNOP method and emphasized the important role of initial errors in creating the SPB for El Ni#cod#241; events. (Peng et al., 2012) also demonstrated the dominant role of initial errors in error growth related to the SPB by investigating the effect on the SPB of model errors associated with a stochastic Madden-Julian Ocsillation (MJO).

From analyses of initial and parameter errors, we know that initial errors can lead to a more noticeable SPB phenomenon and have a greater influence than parameter errors. However, note that these conclusions only apply to cases in which the two types of errors are considered separately without their nonlinear interactions. However, there can be cases in which both initial and parameter errors are present in the model, and the optimal error combination that leads to the greatest prediction errors may induce a more significant SPB than if only initial errors exist in the model. In this case, we cannot overlook the importance of the parameter errors. (Mu et al., 2010) discussed such a case in the WF 96 model. The WF 96 model is a simplified theoretical model and contains only two parameters and two variables. (Yu et al., 2012) attempted to calculate the CNOP-I and CNOP-P errors in the ZC model and also studied a simple combination of the two types of errors. However, this group did not consider the nonlinear interaction between the two types of errors. Therefore, there is no guarantee that a simple combination of the CNOP-I and CNOP-P errors is an optimal case, and in certain cases, this combination may even lead to smaller prediction errors compared with CNOP-I errors because the CNOP-I and CNOP-P errors tend to interact with each other. The difference of the present paper from (Mu et al., 2010) and (Duan and Zhang, 2010) is in the use of an intermediate ZC model. Using the CNOP approach in the ZC model, we aim in the present paper to investigate both the initial errors and parameter errors and to analyze whether the optimal combination of initial and parameter errors is likely to cause a more noticeable SPB phenomenon than the initial errors alone, which is the main difference from (Yu et al., 2012).

The remainder of the paper is organized as follows. In section 2, we briefly introduce the CNOP approach, and then provide an introduction to our experiments in section 3. In section 4, we present the results of the error growth. Section 5 attempts to explain the reasons behind the observed phenomenon. And finally, a discussion and summary are presented in section 6.

We use the CNOP approach (Mu and Duan, 2003; Mu et al., 2010) to seek the optimal combination of errors that can lead to the largest prediction errors under a given constraint. The CNOP approach has been used in studies of predictability problems (Duan and Mu, 2009; Duan et al., 2013b; Qin et al., 2013). The approach is briefly described below.

M represents a numerical model. The model result at time t is U(t), defined as

U(t) =M_t(P)U₀, (1)

where M_t(P)(U₀) denotes integration of the model with parameter P and initial value U₀ to time t. U(t) is also referred to as the basic state. P, U(t), and U₀ are vectors, where P=(P₁,P₂,. . .,P_m) is the model parameter vector, U(t)=(U₁(x,t),U₂(x,t),. . .,U_n(x,t)) consists of n state variables, U₀ is the initial value of U(t), and x=(x₁,x₂,. . .,x_l) represents the spatial coordinate. If there are errors in the initial value and the parameters (u₀,p, respectively), those errors can cause prediction errors. The prediction result at a given time is U(t)+u(u₀,p,t), and thus we have

U(t)+u(u₀,p,t)= M_t(P+p)(U₀+u₀), (2)

where u(u₀,p,t) is the nonlinear growth of the errors at time t caused by u₀,p. We define the cost function J to describe the magnitude of errors at time t as

J(u₀,p) =||u(u₀,p,t)||, (3)

where ||.|| is the norm of a vector. The errors leading to the largest value of J with the constraints #cod#948; and #cod#963; at time t are defined as u_0,#cod#948;,p_#cod#963;, resulting in

where u_0,#cod#948;,p_#cod#963; is also known as the CNOP errors or the optimal error combination we are seeking.

Additionally, we can only consider the initial errors u₀. In other words, the model is assumed to be perfect, without any parameter errors. The cost function and the optimization problem at this time are

and

where u_{0,#cod#948;,I} is CNOP-I, denoted with I to show its difference from u_0,#cod#948; mentioned above.

Similarly, when we consider only the parameter errors, we can write the cost function and the optimization problem as

and

where p_#cod#963;,p is defined as CNOP-P, which can lead to the largest prediction error when only parameter errors exist.

It is clear that u_0,#cod#948;,p_#cod#963; is not necessarily the simple combination of u_{0,#cod#948;,I} and p_#cod#963;,p. Therefore, the CNOP errors may include the nonlinear interaction between errors and may cause a more noticeable SPB phenomenon. The existing optimization solvers are often used to compute minimization problems, while the CNOP errors are related to a constrained maximization problem. To compute the CNOP, CNOP-I, and CNOP-P errors, we transform the corresponding maximization optimization problem mentioned previously into a minimization problem by considering the negative of the cost function. Accordingly, we use the SPG2 method (Birgin et al., 2000, 2001) to calculate the optimal perturbations. The gradient of the cost function is necessary; therefore, we use the adjoint of the corresponding model to obtain the gradient required by the SPG2.

After integrating the ZC model over a period of 100 years, we obtain a series of El Ni#cod#241; events, which have an approximate 4-yr period and tend to peak at the end of each year. In the numerical experiment, we choose eight events, including both strong and weak El Ni#cod#241; events. For each of these events, we make predictions based on a 12-month lead-time and with start times that coincide with eight different initial months. In this work, we use year (0) to denote the year in which El Ni#cod#241; attains a peak value. Year (-1) and year (1) denote the year before and the year after year (0), respectively. In the numerical experiment, the eight different initial months are July (-1), October (-1), January (0), April (0), July (0), October (0), January (1), and April (1). We refer to the predictions starting from the first four months as growing-phase predictions because those predictions start in the previous seasons that extend through boreal spring in the growing phase of El Ni#cod#241;. Similarly, the predictions starting from the last four months are referred to as the decaying-growth predictions because they cover boreal spring during the decaying phase of El Ni#cod#241;. In total, we choose eight El Ni#cod#241; events, each with eight initial months for a total of 64 cases.

The initial error u₀ consists of two parts——errors from the sea surface temperature anomaly, i.e. SSTA, and the thermocline depth anomaly. We define u₀=(w₁T'₀,w₂h'₀), where w₁=(2^#cod#x000b0; C)^-1 and w₂=(50 m)^-1 are the characteristic scales of the SST and the thermocline depth anomaly, respectively. The constrain condition is |u₀| 1, and the norm is

where T'_0,(i,j) and h'_0,(i,j) are the SSTA and thermocline depth anomalies at different grid points, respectively, and (i,j) is the grid point in the domain of the tropical Pacific from 129.375^#cod#x000b0;E to 84.375^#cod#x000b0;W (at an interval of 5.625^#cod#x000b0;) and 19^#cod#x000b0;S to 19^#cod#x000b0;N (at an interval of 2^#cod#x000b0;).

According to (Yu et al., 2012), we choose nine main empirical parameters, which are listed in Table 1, including their physical meanings and reference values [for details, please see (Zebiak and Cane, 1987) and (Yu et al., 2012)]. The parameter error vector is p=(p₁,p₂,. . .,p₉). The constraint condition is p||p_i|#cod#8804;#cod#963;_i,i=1,2,#cod#8230;,9, where #cod#963;_i=x_iP_i/100, and the value of x_i is listed in Table 1 (right-hand column). To satisfy this requirement, the parameter errors must satisfy the constraint condition -x_iP_i/100#cod#8804; p_i#cod#8804; x_iP_i/100. For example, the reference value of parameter P₁ is 1.6, and the error bound is x₁=0.1, as shown in Table 1; therefore, the interval of p₁ is [-1.6#cod#215; 0.1%,1.6#cod#215; 0.1%]. The constraint bounds are defined according to the basic rule that, in realistic ENSO prediction, the forecast model should first guarantee that it could simulate the main features of the observed ENSO. As such, the values of the parameters in the model must be set to satisfy this precondition.

The definition of the cost function J(u₀,p) is described as

where w₁ has the same meaning as mentioned previously, and T'(u₀,p,t) represents the prediction errors of the SSTA at the prediction time t caused by both initial error u₀ and parameter error p. The purpose of the present paper is to investigate whether the CNOP errors can cause a more noticeable SPB phenomenon. To compare the growth of different types of errors, we also calculate u_{0,#cod#948;,I} (CNOP-I errors) and p_#cod#963;,p (CNOP-P errors), as well as the simple combination of these two types of errors (CNOP-I + CNOP-P errors).

From the perspective of error growth, the SPB can be understood as the error growth in spring being faster than that in the other seasons. Therefore, in this section, we investigate whether the CNOP errors could be the cause of the fastest error growth in spring.

We divide a calendar year into four seasons: January-March (JFM), April-June (AMJ), July-September (JAS), and October-December (OND). Here we also define AMJ as spring, which is consistent with (Yu et al., 2012). We take the first El Ni#cod#241; event beginning in July (-1) as an example to illustrate how to calculate the error growth tendency. We integrate the ZC model with each of the CNOP, CNOP-I, CNOP-P, and CNOP-I + CNOP-P errors. As a result, we obtain ||T′(u_0,#cod#948;,p_#cod#963;, t)||,||T′(u_{0,#cod#948;,I}, t)||,||T′(p_#cod#963;,p, t)||, and ||T′(u_{0,#cod#948;,I},p_#cod#963;,p, t)||, where ||T′(u_0,#cod#948;,p_#cod#963;,t)|| is the norm of the prediction errors induced by u_0,#cod#948;,p_#cod#963;, and t=1,2,. . .,13. t=1 denotes July (-1) and t=13 denotes July (0). Prediction error is defined as:

The error growth tendency #cod#x003ba; is

where t=1,2,. . .,12 and the seasonal error growth tendency is the seasonal mean of #cod#x003ba;. Similarly, we can calculate the seasonal error growth tendencies of different El Ni#cod#241; events starting from other initial months.

Figure 1 shows the resulting seasonal growth tendencies of the four types of errors in the prediction experiments starting from July (-1) (Fig. 1a), October (-1) (Fig. 1b), January (0) (Fig. 1c), and April (0) (Fig. 1d) for the eight El Ni#cod#241; events. The bars indicate the mean values of the eight El Ni#cod#241; events, and the error bars on each bar display the minimum and maximum values of the eight El Ni#cod#241; events. We compare the ensemble means of the seasonal error growth tendencies in different seasons and find that, for prediction months starting from July (-1) and October (-1), the CNOP errors show the most noticeable growth tendency in spring (AMJ). As for Fig. 1c, the largest seasonal error growth tendency of the CNOP errors occurs in JAS. We also find that AMJ still has a noticeable error increase. For the predictions starting from April (0), which occur in the spring, the SPB phenomenon is not noticeable. The CNOP-I and CNOP-I + CNOP-P errors show similar results. In contrast, the seasonal error growth tendency caused by the CNOP-P errors is much smaller, which means that the CNOP-P errors do not lead to a noticeable SPB. For the decaying-phase predictions, which are not shown in this paper, we obtain a similar result to that of growing-phase predictions.

Figure 1.  Ensemble means of the seasonal error growth tendencies for eight El Ni#cod#241; events starting from Jul (-1), Oct (-1), Jan (0), and Apr (0) caused by the CNOP-I (green bars), CNOP-P (black bars), CNOP-I + CNOP-P (red bars), and CNOP errors (blue bars). The bar value denotes the ensemble mean of the eight El Ni#cod#241; events, and the error bars superimposed on each bar indicate the minimum and maximum values of the eight El Ni#cod#241; events.

It should be noted that the error growth tendencies induced by the CNOP errors are not much larger than those caused by the CNOP-I errors, which may imply that the CNOP errors do not lead to a more noticeable SPB phenomenon than the CNOP-I errors.

The definition of the prediction errors is given by the values of ||T′(u_0,#cod#948;,p_#cod#963;, t)||,||T′(u_{0,#cod#948;,I}, t)||,||T′(p_#cod#963;,p, t)||, and ||T′(u_{0,#cod#948;,I},p_#cod#963;,p, t)||, where ||T′(u_0,#cod#948;,p_#cod#963;,t)|| at t=13, which are the SSTA errors induced by the CNOP, CNOP-I, CNOP-P, and CNOP-I + CNOP-P errors, respectively.

Figure 2.  The growing-phase SSTA prediction errors caused by the CNOP-I, CNOP-P, CNOP-I + CNOP-P, and CNOP errors for the eight El Ni#cod#241; events. The numbers 1-8 on the horizontal axis denote the eight El Ni#cod#241; events.

Figure 2 shows the growing-phase prediction errors for the eight El Ni#cod#241; events. The result of the decaying-phase predictions is similar to that of the growing-phase predictions, which are not shown. Figure 2 illustrates that the prediction errors caused by the CNOP errors are just slightly higher than those caused by the CNOP-I errors for different El Ni#cod#241; events and different initial months. Therefore, we conclude that the CNOP errors do not lead to a more noticeable SPB phenomenon compared with the CNOP-I errors.

Note that the CNOP-P errors produce notably small prediction errors, which means that parameter errors are less important than the initial errors. In addition, the prediction errors induced by the linear combined mode, the CNOP-I + CNOP-P errors, are sometimes larger and sometimes smaller than those caused by the CNOP-I errors. These results are most likely due to the fact that the initial errors and the parameter errors occasionally restrain each other such that the result is less than it would be if initial errors exist alone, and in the other case the two types of errors mutually stimulate each other such that the resulting prediction error caused by the CNOP-I + CNOP-P errors is larger.

In this section, we investigate the evolution of the SSTA prediction errors. We choose four different months after the starting month——namely, t=1,4,7,10——where t=1 denotes the starting month, t=4 denotes three months after the starting month, and so on. As shown in Fig. 1, predictions made from October (-1) show the most notable SPB. So, we take the third El Ni#cod#241; event with an initial month of October (-1) as an example to compare the patterns of different prediction errors of the SSTA.

Figure 3.  The evolution of SSTA (^#cod#x000b0;C) caused by the CNOP-I (the first column), CNOP-P (the second column), CNOP-I + CNOP-P (the third column), and CNOP (the last column) errors for the third El Ni#cod#241; event starting from Oct (-1) in the tropical Pacific.

Figure 3 shows the resulting error growth patterns of the SSTA. We observe that at the initial time, the CNOP errors are similar to the CNOP-I errors, with a positive value in the eastern tropical Pacific and a negative one to the west. No initial SSTA errors exist at the initial time for the CNOP-P errors, a case in which the initial error is not considered. With the progression of time, the prediction error caused by the CNOP errors becomes similar to that caused by CNOP-I errors, not only in the structure but also in the amplitude. The prediction error caused by the CNOP-P errors is similar to other prediction errors in the pattern, but with different amplitude. For other prediction experiments, we come to a similar conclusion. Therefore, we have demonstrated that the CNOP errors do not lead to a more significant SPB than the CNOP-I errors. In other words, the initial errors are the critical factor and are more important than the parameter errors for the SPB in the ZC model.

The previous sections have shown that the CNOP errors do not lead to a more noticeable SPB than the CNOP-I errors. In this section, we attempt to explain this result. Previous studies have shown that the SPB may result from the combined effects of the annual cycle, El Ni#cod#241; events and particular initial errors in which the Bjerknes feedback (Bjerknes, 1969) could account for the growth of initial errors related to the SPB (Yu et al., 2009). According to the Bjerknes feedback, the SSTA, cold water upwelling, and the wind stress anomaly are crucial variables. Therefore, we investigate these three variables to identify how the CNOP errors affect these variables, and we begin by asking the question "what are the differences among the three variables affected by the CNOP errors and the CNOP-I errors?"

We design two experiments: Experiment One is superimposed with the CNOP-I errors and Experiment Two is superimposed with the CNOP errors. Next, we integrate the ZC model and obtain the evolutions of the three variables in each experiment. The purpose is to identify the differences between the two experiments. Figure 4 shows the differences in the three variables between the two experiments: SSTA (left column), wind (middle column), and upwelling velocity (right column). From Fig. 4 we note that the differences are not large. Therefore, this result demonstrates that the CNOP errors do not have more significant effects on the three variables than the CNOP-I errors. We deduce that the CNOP errors do not affect the error growth more significantly than the CNOP-I errors.

Figure 4.  The differences of the evolution of SSTA (left column), wind (middle column), and upwelling velocity (right column) as affected by the CNOP and CNOP-I errors; this is the result of the third El Ni#cod#241; event with the initial month of Oct (-1).

Previous studies have shown that ENSO is a nonlinear oscillation system and the nonlinearity causes the amplitude asymmetry of ENSO (An and Jin, 2004; Duan et al., 2004; Rodgers et al., 2004). Additionally, (Duan et al., 2009a) demonstrated that the nonlinearity suppresses the amplitude of El Ni#cod#241; during the decaying phase and favors the decaying of El Ni#cod#241; events. The nonlinear temperature advection (NTA), sub-surface temperature parameterization (STP), and wind stress anomalies (WSA) are three crucial nonlinear terms in the ZC model that can influence the evolution of ENSO. Recently, (Duan et al., 2013a) investigated how these three types of nonlinear terms affect the evolution of ENSO and illustrated that the nonlinearities originating from NTA enhance El Ni#cod#241; events, whereas the nonlinearities that originate from STP and WSA suppress El Ni#cod#241; events. Furthermore, the enhancement effect of the NTA is larger than the suppression effect of the STP and WSA. Although we have shown above that the net effects of the CNOP errors and CNOP-I errors are similar, there is a possibility that the nonlinear terms in the ZC model have changed but that the net effect is balanced by the three nonlinear terms. For example, the CNOP errors may lead to a larger NTA and a smaller STP and WSA than the CNOP-I errors, but the two types of errors may still produce a similar result. Therefore, in this section, we investigate the nonlinear terms in the ZC model. Similarly, we also choose the NTA, STP and WSA, and assess how the CNOP errors affect these terms (for details of the three nonlinear terms, please see the appendix).

To illustrate the different effects between the CNOP errors and the CNOP-I errors on the three nonlinear terms, we superimpose the ZC model with the CNOP errors and the CNOP-I errors separately. Next, we integrate the model for one year to investigate the developments of the nonlinear terms. In Fig. 5, we find that the results of the two types of errors are rather similar. For other El Ni#cod#241; events and initial times, we obtain similar results (details omitted here). Therefore, we conclude that neither the Bjerknes feedback nor the internal nonlinear processes are significantly changed.

In summary, the evolutions of the three nonlinear terms when the CNOP errors are superimposed in the model are similar to those when the CNOP-I errors are superimposed, indicating that the CNOP errors do not have much influence on the nonlinear processes in the ZC model. Additionally, the Bjerknes feedback does not change when superimposed with the CNOP errors. These two points imply that the mechanism of error growth will not display a large change. Additionally, the initial components of the CNOP errors (i.e., u_0,#cod#948;) and the CNOP-I errors (i.e., u_{0,#cod#948;,I}) are nearly the same, not only in their quantities but also in the patterns displayed (figures omitted here). Therefore, the prediction errors are quite similar, thus demonstrating why the CNOP errors do not contribute to a more notable SPB phenomenon than the CNOP-I errors.

Figure 5.  The evolution of nonlinear temperature advection (NTA), sub-surface temperature parameterization (STP), and wind stress anomalies (WSA) superimposed with the CNOP errors (solid line) and CNOP-I errors (dashed line); this is the result of the third El Ni#cod#241; event with the initial month of Oct (-1).

In this paper, we applied the approach of conditional nonlinear optimal perturbation (CNOP) to the ZC model to study the effect of initial errors and parameter errors on El Ni#cod#241; predictability and to identify the errors playing a vital role in generating a significant SPB for El Ni#cod#241; events. We chose eight different El Ni#cod#241; events and eight different initial months for each event, and we calculated the CNOP (optimal error combination), CNOP-I (optimal initial errors when only initial errors are considered), CNOP-P (optimal parameter errors when only parameter errors exist), and CNOP-I + CNOP-P (simple combination of the CNOP-I errors and the CNOP-P errors) errors with reasonable constraint conditions. By comparing the error evolution, we concluded that: (1) from the perspective of the seasonal growth tendencies of the errors, the CNOP errors show an obvious season-dependent evolution, which is not significantly greater than that of the CNOP-I errors; (2) from the view-point of the prediction errors, although the CNOP errors may lead to the largest prediction errors, the CNOP errors are not significantly larger than the prediction errors caused by the CNOP-I errors; and (3) the patterns of error evolutions show that the CNOP errors and the CNOP-I errors are notably similar to each other in terms of not only their structures but also their amplitudes. All of the findings listed above support the same conclusion: the CNOP errors do not cause a more noticeable SPB than the CNOP-I errors. In other words, although the initial and parameter errors coexist in the ZC model, the parameter errors do not stimulate the evolution of the initial errors or lead to a much more significant SPB phenomenon compared with the CNOP-I errors. This result may imply that the nonlinear interaction between the two types of errors is not as strong as expected. Previous studies have demonstrated a minor role of parameter errors in yielding the SPB for ENSO events using the ZC model and CNOP-I and CNOP-P (Yu et al., 2012), or using CNOP and a theoretical model (Duan and Zhang, 2010). In the present paper we used the ZC model and CNOP to further demonstrate the minor role of parameter errors. Therefore, combining these three studies, we can conclude that the parameter errors indeed play a minor role in yielding the SPB for ENSO events in certain models. The results may provide an idea for improving the prediction ability of ENSO during spring by simply focusing on reducing the initial errors. In section 5, we discussed the reasons why the CNOP errors do not contribute to a more noticeable SPB phenomenon than the CNOP-I errors. We found that the Bjerknes feedback and the internal nonlinear processes are almost the same when superimposed with CNOP and CNOP-I errors, respectively. In addition, the initial components of the CNOP errors (i.e., u_0,#cod#948;) and the CNOP-I errors (i.e., u_{0,#cod#948;,I}) are quite similar, and therefore the ultimate results are not significantly different.

In the present work, one of the critical factors in obtaining our conclusions is the selection of constraint conditions, which requires further discussion. Realistic analysis errors in tropical areas are 0.2^#cod#x000b0;C for the SSTA (Kaplan et al., 1998) and 15 m for the thermocline depth anomaly (Karspeck et al., 2006). Our results showed that the SSTA is approximately 0.01^#cod#x000b0;C and the maximum thermocline depth anomaly is approximately 3 m. Therefore, the magnitude of the CNOP-I errors is smaller than that of realistic analysis errors. However, the parameter errors used here are the largest allowed for ENSO simulation. From this point of view, our results are believable to a certain extent.

Although the ZC model is only a simplified model, it is still used for real-time ENSO prediction. Therefore, our results may provide a new idea to improve ENSO prediction in the ZC model. As for other models, further studies are required. Because the main characteristics of La Ni#cod#241; events are poorly simulated by the ZC model (An and Wang, 2001), we have not investigated the corresponding problem for La Ni#cod#241; events, a scenario that should be analyzed with other models. In addition, many other factors may affect the final prediction results, such as various schemes of physical parameterizations, atmospheric noise, or other high frequency variations (i.e., westerly wind bursts). The extent to which these factors affect the SPB phenomenon should be studied in future research.

The nonlinear temperature advection (NTA) is expressed as

where

and are the mean horizontal currents and upwelling, respectively, and H₁=50 m. Here we use bolded simbles to represent vectors or matrix and unbolded simbles are scalars. is the prescribed mean SST, and is the prescribed mean vertical temperature gradient. The entrainment temperature anomaly, T _e, is defined by

T_e=#cod#947;T_sub+(1−#cod#947;)T, (A3)

where T_sub is the sub-surface temperature parameterization (STP), #cod#947;=0.75.

The sub-surface temperature parameterization (STP) is

where is the prescribed mean upper layer depth and T₁,b₁,T₂,b₂ are listed in Table 1.

The wind stress anomalies (WSA) is

where #cod#961;₀C_D=0.0329 (i.e., #cod#963; in Table 1), , and V=(u,v). For additional details, please see (Zebiak and Cane, 1987).